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Finding Remainder of Numbers of the Form a^b

If 24n+4 - 15...

Question

If $2^{4 n + 4} - 15 n - 16, n \in N$ is divided by $225$ , then the remainder is

A

0

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B

224

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C

1

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D

15

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Solution

The correct option is A $0$
Given:
$2^{4 n + 4} - 15 n - 16 = 2^{4 (n + 1)} - 15 n - 16 = 16^{n + 1} - 15 n - 16 = (1 + 15)^{n + 1} - 15 n - 16 = 1 +^{n + 1} C_{1} \cdot 15 +^{n + 1} C_{2} \cdot 15^{2} + \dots +^{n + 1} C_{n + 1} \cdot (15)^{n + 1} - 15 n - 16 = 1 + 15 (n + 1) + 15^{2} [k] - 15 n - 16 = 225 k, k \in N$

Hence, the remainder is $0$ .

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2^{4 n + 4} - 15 n - 16, n \in N

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Finding Remainder of Numbers of the Form a^b

Standard XIII Mathematics

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